68,
I was thinking that I really owe your thread an apology. My post about inverter/alternator had nothing to do with your original question. The reason I hadn't chipped in initially was you were looking for something relatively quick and inexpensive, which isn't my battery bailiwick.
But now that you are asking about AGM (even though I'm not kdarling), no, there isn't really anything wrong with putting AGM's outside. It's the opposite: You CAN put them inside less inconveniently than flooded cells (due to lesser venting requirements plus the ability to orient them numerous ways). They don't love heat (most batteries don't), but unless you are in a very hot area (and keep your rig air-conditioned all the time), inside probably won't be that much cooler than outside so I don't see that as a big thing.
Most batteries don't mind cold, although IIRC they may be slightly less efficient while they are cold (but MUCH happier than if they are kept in a hot place). Basically, back in the day when boats all had flooded cells, winter storage could be just charging them up and leaving them (very little self-discharge at cold temps). So flooded cells, outside usually (but could go inside with some design considerations); AGM can also go outside but easier to put inside than flooded cells.
With lithiums there is a bit more impetus to put them inside (vs. either/or), because they can't be charged below freezing (and they don't like heat). They can be discharged below freezing though. Probably not on your menu though, because if you want to make AGM's look cheap, price lithiums

(they do pencil out pretty well in the long run, but that's dependent on use case and other factors, plus you have to buy them initially).
I see a few advantages to lithium, depending on how you use your bank:
1) They don't have the lengthy absorb stage. Not such a big deal if you are plugged in (almost unnoticeable), but if you are running on solar power, then giving up 2-3 hours of prime sun time to stuff wee bits of power through the eye of a needle is a sacrifice.
2) They don't have the big Peukert penalty. This is of no importance to me as I don't tend to run that type of loads; but things like microwaves, Air-con, and induction cooktops make a difference.
3) They don't mind not getting up to 100% charge, and you can draw them down further.
4) Lastly, they are lighter weight (maybe not such a big deal in a 10,000# RV unless you plan to stuff a large number of batteries in).
If I had to replace my batteries right now, with my boondocking/solar use case... I'd consider lithium but not sure I'd go with them. They'd be nice, but I am already able to get up to 100% most days, I don't tend to run big Peukert loads, and I am still way below my GVWR. Since my AGM bank is still healthy, I can put off any decision (and meanwhile things are still being learned about how best to treat them, charge them, etc.). Not that I'd discourage anyone if they wanted them now.
To me the big benefit of going AGM (this was back in 2014) was being able to bring them inside. It made the wiring easier/shorter, and I find the vented/outdoor storage SO valuable (where the batteries used to be). And the lack of outgassing/corrosion is nice. But that's an individual thing so others would likely feel differently.
BG
PS: kdarling, that's interesting about Trojan making lithium now. I am comparing different lithium options for another project wherein weight is super important (and I don't have room for more than two batteries or much solar so space/no-absorption is too). The Battle Borns look interesting, and they were really nice when I called; but I'm SO SPOILED by the Lifeline battery manual! Even some of the more expensive lithiums don't have the equivalent literature. If Trojan is going to put out better documentation, fantastic.